57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 1 Chapter 7 Dimensional Analysis and Modeling
The Need for Dimensional Analysis
Dimensional analysis is a process of formulating fluid mechanics problems in terms of nondimensional variables and parameters.
1. Reduction in Variables: F = functional form If F(A1, A2, …, An) = 0, Ai = dimensional variables
Then f(1, 2, … r < n) = 0 j = nondimensional parameters
Thereby reduces number of = j (Ai) experiments and/or simulations i.e., j consists of required to determine f vs. F nondimensional groupings of Ai’s
2. Helps in understanding physics
3. Useful in data analysis and modeling
4. Fundamental to concept of similarity and model testing
Enables scaling for different physical dimensions and fluid properties
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 2 Dimensions and Equations
Basic dimensions: F, L, and t or M, L, and t F and M related by F = Ma = MLT-2
Buckingham Theorem
In a physical problem including n dimensional variables in which there are m dimensions, the variables can be arranged into r = n – mˆ independent nondimensional parameters r (where usually mˆ = m).
F(A1, A2, …, An) = 0
f(1, 2, … r) = 0
Ai’s = dimensional variables required to formulate problem (i = 1, n)
j’s = nondimensional parameters consisting of groupings of Ai’s (j = 1, r)
F, f represents functional relationships between An’s and r’s, respectively
= rank of dimensional matrix = m (i.e., number of dimensions) usually
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 3 Dimensional Analysis
Methods for determining i’s
1. Functional Relationship Method
Identify functional relationships F(Ai) and f(j)by first determining Ai’s and then evaluating j’s a. Inspection intuition b. Step-by-step Method text c. Exponent Method class
2. Nondimensionalize governing differential equations and initial and boundary conditions
Select appropriate quantities for nondimensionalizing the GDE, IC, and BC e.g. for M, L, and t
Put GDE, IC, and BC in nondimensional form
Identify j’s
Exponent Method for Determining j’s
1) determine the n essential quantities 2) select mˆ of the A quantities, with different dimensions, that contain among them the dimensions, and use them as repeating variables together with one of the other A quantities to determine each . 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 4
For example let A1, A2, and A3 contain M, L, and t (not necessarily in each one, but collectively); then the j parameters are formed as follows:
x y z Determine exponents A 1 A 1 A 1 A 1 1 2 3 4 such that i’s are
x2 y2 z2 dimensionless 2 A1 A2 A3 A5
xnm ynm znm 3 equations and 3 nm A1 A2 A3 An unknowns for each i
In these equations the exponents are determined so that each is dimensionless. This is accomplished by substituting the dimensions for each of the Ai in the equations and equating the sum of the exponents of M, L, and t each to zero. This produces three equations in three unknowns (x, y, t) for each parameter.
In using the above method, the designation of mˆ = m as the number of basic dimensions needed to express the n variables dimensionally is not always correct. The correct value for mˆ is the rank of the dimensional matrix, i.e., the next smaller square subgroup with a nonzero determinant.
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 5
Dimensional matrix = A1 ……… An M a11 ……… a1n L aij = exponent t a31 ……… a3n of M, L, or t in o ……… o Ai : : : : : : o ……… o
n x n matrix
Rank of dimensional matrix equals size of next smaller sub-group with nonzero determinant
Example: Hydraulic jump (see section 15.2)
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 6
Say we assume that
V1 = V1(, g, , y1, y2) or V2 = V1y1/y2
Dimensional analysis is a procedure whereby the functional relationship can be expressed in terms of r nondimensional parameters in which r < n = number of variables. Such a reduction is significant since in an experimental or numerical investigation a reduced number of experiments or calculations is extremely beneficial
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 7
1) , g fixed; vary Represents 2) , fixed; vary g many, many 3) , g fixed; vary experiments
In general: F(A1, A2, …, An) = 0 dimensional form
f(1, 2, … r) = 0 nondimensional form with reduced
or 1 = 1 (2, …, r) # of variables
It can be shown that
V1 y2 Fr Fr gy 1 y1 neglect ( drops out as will be shown) thus only need one experiment to determine the functional relationship
2 1 2 Fr x x x Fr 2 0 0 1/ 2 1 ½ .61 Fr x1 x 2 1 1 2 1.7 5 3.9
For this particular application we can determine the functional relationship through the use of a control volume analysis: (neglecting and bottom friction) x-momentum equation: Fx VxVA 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 8 y2 y2 1 2 V V y V V y Note: each term 2 2 1 1 1 2 2 2 in equation must have some units: 2 2 2 2 y1 y2 V2 y2 V1 y1 principle of 2 g dimensional homogeneity, i.e., in this case, continuity equation: V1y1 = V2y2 force per unit width N/m
V1y1 V2 y2
2 y2 y y 1 1 2 V2 y 1 1 2 y 1 g 1 y 1 2 pressure forces = inertial forces due to gravity y 1 2 y3 y 1 now divide equation by 1 gy 2 2 V1 1 y2 y2 1 dimensionless equation gy 1 2 y1 y1 ratio of inertia forces/gravity forces = (Froude number)2 note: Fr = Fr(y2/y1) do not need to know both y2 and y1, only ratio to get Fr Also, shows in an experiment it is not necessary to vary
, y1, y2, V1, and V2, but only Fr and y2/y1 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 9
Next, can get an estimate of hL from the energy equation (along free surface from 12)
V2 V2 1 y 2 y h 2g 1 2g 2 L
3 y2 y1 h L 4y1y2
f() due to assumptions made in deriving 1-D steady flow energy equations
Exponent method to determine j’s for Hydraulic jump
F(g,V1,y1,y2,,) = 0 n = 6 use V1, y1, as L L M M L L 2 T 3 LT repeating variables T L Assume mˆ = m to avoid evaluating x1 y1 z1 m = 3 r = n – m = 3 rank of 6 x 6 1 = V1 y1 dimensional matrix = (LT-1)x1 (L)y1 (ML-3)z1 ML-1T-1
L x1 + y1 3z1 1 = 0 y1 = 3z1 + 1 x1 = -1 T -x1 1 = 0 x1 = -1 M z1 + 1 = 0 z1 = -1
1 y1V1 1 or 1 = Reynolds number = Re y1V1
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 10 x2 y2 z2 2 = V1 y1 g = (LT-1)x2 (L)y2 (ML-3)z2 LT-2
L x2 + y2 3z2 + 1 = 0 y2 = 1 x2 = 1 T -x2 2 = 0 x2 = -2 M z2 = 0 gy V V 2 y g 1 1/ 2 1 2 1 1 2 2 = Froude number = Fr V1 gy1
x3 y3 z3 3 = V1 y1 y2 = (LT-1)x3 (L)y3 (ML-3)z3 L
L x3 + y3 3z3 + 1 = 0 y3 = 1 T -x3 = 0 M -3z3 = 0 y2 1 y1 3 3 = depth ratio y1 y2 f(1, 2, 3) = 0 or, 2 = 2(1, 3) i.e., Fr = Fr(Re, y2/y1) if we neglect then Re drops out
V1 y2 Fr f gy 1 y1
Note that dimensional analysis does not provide the actual functional relationship. Recall that previously we used control volume analysis to derive
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 11 2 V1 1 y2 y2 1 gy 1 2 y1 y1 the actual relationship between F vs. y2/y1
F = F(Re, Fr, y1/y2) or Fr = Fr(Re, y1/y2) dimensional matrix:
g V1 y1 y2 M 0 0 0 0 1 1 L 1 1 1 1 3 -1 Size of next smaller t -2 -1 0 0 0 -1 subgroup with nonzero 0 0 0 0 0 0 determinant = 3 = rank 0 0 0 0 0 0 of matrix 0 0 0 0 0 0 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 12 Common Dimensionless Parameters for Fluid Flow Problems Most common physical quantities of importance in fluid flow problems are: (without heat transfer) 1 2 3 4 5 6 7 8 V, , g, , , K, p, L velocity density gravity viscosity surface compressibility pressure length tension change
n = 8 m = 3 5 dimensionless parameters
⁄ VL inertia forces V2 / L 1) Reynolds number = 2 Re
viscous forces V / L
⁄ Rcrit distinguishes among flow regions: laminar or turbulent value varies depending upon flow situation
V inertia forces V2 / L 2) Froude number = Fr gL gravity force
important parameter in free-surface flows
V2L inertia force V2 / L 3) Weber number = We surface tension force / L2
important parameter at gas-liquid or liquid-liquid interfaces ⁄ and when these surfaces are in contact with a boundary
V V inertia force 4) Mach number = Ma k / a compressibility force
speed of sound in liquid ⁄ Paramount importance in high speed flow (V > c)
p pressure force p / L 5) Pressure Coefficient = Cp V2 inertia force V2 / L
(Euler Number) ⁄ 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 13 Nondimensionalization of the Basic Equation
It is very useful and instructive to nondimensionalize the basic equations and boundary conditions. Consider the situation for and constant and for flow with a free surface
Continuity: V 0
DV Momentum: p z 2 V Dt g = specific weight Boundary Conditions: 1) fixed solid surface: V 0 2) inlet or outlet: V = Vo p = po 3) free surface: w p p R 1 R 1 t a x y (z = ) surface tension
All variables are now nondimensionalized in terms of and
U = reference velocity
L = reference length
V tU V* t* U L x p gz x* p* L U2 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 14 All equations can be put in nondimensional form by making the substitution V V*U
t* U t t* t L t*
ˆi ˆj kˆ x y z x* y* z* ˆi ˆj kˆ x* x y* y z* z 1 * L
u 1 U u* and Uu* etc. x L x* L x*
Result: * V* 0 DV* *p* *2 V* Dt VL 1) V* 0 Re-1 V p 2) V* o p* o U V2 * * * po gL * *1 *1 3) w p z R x R y t* V2 U2 V2L pressure coefficient -1 Fr-2 We V = U 57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 15 Similarity and Model Testing
Flow conditions for a model test are completely similar if all relevant dimensionless parameters have the same corresponding values for model and prototype
i model = i prototype i = 1, r = n - mˆ (m)
Enables extrapolation from model to full scale
However, complete similarity usually not possible
Therefore, often it is necessary to use Re, or Fr, or Ma scaling, i.e., select most important and accommodate others as best possible
Types of Similarity:
1) Geometric Similarity (similar length scales): A model and prototype are geometrically similar if and only if all body dimensions in all three coordinates have the same linear-scale ratios
= Lm/Lp ( < 1)
1/10 or 1/50 2) Kinematic Similarity (similar length and time scales): The motions of two systems are kinematically similar if homologous (same relative position) particles lie at homologous points at homologous times
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 16 3) Dynamic Similarity (similar length, time and force (or mass) scales): in addition to the requirements for kinematic similarity the model and prototype forces must be in a constant ratio
Model Testing in Water (with a free surface)
F(D, L, V, g, , ) = 0
n = 6 and m = 3 thus r = n – m = 3 pi terms
In a dimensionless form,
f(CD, Fr, Re) = 0 or CD = f(Fr, Re) where
√
Vm Vp If Frm Frp or gL m gL p
Vm Vp Froude scaling
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 17
VmLm VpLp and Rem = Rep or m p VL m m m 3/ 2
pVL p p Then,
or
However, impossible to achieve, since 8 2 7 2 if 1 10 , m 3.1 10m s 1.2 10 m s 72 For mercury 1.2 10 ms
Alternatively one could maintain Re similarity and obtain
Vm = Vp/
But, if , VVmp10 ,
High speed testing is difficult and expensive.
2 V2 V m p gmLm gpLp
2 gm Vm Lp 2 gp Vp Lm
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 18 2 gm Vm Lp 2 gp Vp Lm
g 1 1 m 3 2 gp
gp gm 3
But if 1 10 , ggmp1000 Impossible to achieve
Model Testing in Air
F(D, L, V, , , a) = 0
n = 6 and m = 3 thus r = n – m = 3 pi terms
In a dimensionless form,
f(CD, Re, Ma) = 0 or CD = f(Re, Ma) where
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 19 V L V L If m m p p m p
V V and m p a m a p
Then,
or
1 L a However, m m m p Lp a p again not possible
Therefore, in wind tunnel testing Re scaling is also violated
57:020 Mechanics of Fluids and Transport Processes Chapter 7 Professor Fred Stern Fall 2013 20 Model Studies w/o free surface 1 c p V2 p 2 See High Re text Model Studies with free surface
In hydraulics model studies, Fr scaling used, but lack of We similarity can cause problems. Therefore, often models are distorted, i.e. vertical scale is increased by 10 or more compared to horizontal scale
Ship model testing:
CT = f(Re, Fr) = Cw(Fr) + Cv(Re)
Cwm = CTm Cv Vm determined Based on flat plate of for Fr scaling same surface area CTs = Cwm + Cv